Functional stochastic classical mechanics

L. Boltzmann, Lectures on Gas Theory (Univ. California Press, Berkeley, CA, 1964).

A. Poincaré, “Le mecanisme et l’experience”, Revue de Metaphysique et de Morale 1, 534–537 (1893); English translation, Stephen Brush, Kinetic Theory, 2, p.203.

N. N. Bogolyubov, Problems of Dynamic Theory in Statistical Physics (OGIZ, 1946) [in Russian].

N. S. Krylov,Works on the Foundations of Statistical Physics (Akad. Nauk. SSSR, Moscow-Leningrad, 1950) [in Russian].

L. D. Landau and E.M. Lifshitz, Statistical Physics, part 1 (3 ed., Pergamon, 1980).

G. V. Chester, “The theory of irreversible processes,” Rep. Progr. Phys. 26(1), 411–472 (1963).

M. Kac, Probability and Related Topics in Physical Sciences (JohnWiley, New York, 1959).

D. N. Zubarev, Nonequilibrium Statistical Thermodynamics (Consultants Bureau, New York, 1974).

M. Ohya, “Some aspects of quantum information theory and their applications to irreversible processes” Rep. Math. Phys. 27, 19–47 (1989).

I. Prigogine, Les Lois du Chaos (Flammarion, Paris, 1994).

V. V. Kozlov, Temperature Equilibrium on Gibbs and Poincaré (Moscow-Ijevsk, 2002) [in Russian].

V. V. Kozlov, Gibbs Ensembles and Nonequilibrium Statistical Mechanics (Moscow-Ijevsk, 2008).

L. Accardi, Y. G. Lu and I. V. Volovich, Quantum Theory and Its Stochastic Limit (Springer, Berlin, 2002).

V. L. Ginzburg, “Which problems in physics are most important and interesting in the beginning of XXI century?” In: V. L. Ginzburg, On Science, on Myself and Others, pp. 11–74, (Fizmatlit, 2003) [in Russian].

G. Gallavotti, “Fluctuation relation, fluctuation theorem, thermostats and entropy creation in non equilibrium statistical physics,” [arXiv:cond-mat/0612061] (2006).

R. Feynman, The Character of Physical Law, A series of lectures at Cornell University, USA (Cox and Wyman Ltd., London, 1965).

J. L. Lebowitz, “From time-symmetric microscopic dynamics to time-asymmetric macroscopic behavior: An overview,” [arXiv:0709.0724] (2007).

S. Goldstein, “Boltzmann’s approach to statistical mechanics,” in: J. Bricmont et al. (Eds), Chance in Physics: Foundations and Perspectives, Lecture Notes in Physics 574, pp. 39–68 (Springer, Berlin, 2001).

J. Bricmont, “Science of chaos or chaos in science?” in: The Flight from Science and Reason, Annals N.Y. Acad. Sci. 775, 131–182 (1996).

A. N. Kolmogorov, “The general theory of dynamical systems and classical mechanics,” Proc. Int. Congress of Mathematicians (Amsterdam, 1954), 1, 315–333 (North Holland, Amsterdam, 1957) [inRussian]. English translation as Appendix D in R.H. Abraham, Foundations of Mechanics, pp. 263–279 (Benjamin, 1967). Reprinted as Appendix in R.H. Abraham and J.E. Marsden, Foundations of Mechanics, Second ed., pp. 741–757 (Benjamin, Cummings 1978).

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1978).

B. A. Dubrovin, A. T Fomenko and S. P. Novikov, Modern Geometry: Methods and Applications, GTM 93, Part 1 (Springer-Verlag, 1984).

D. V. Anosov, “Geodesic flows on closed Riemann manifolds with negative curvature,” Proc. Steklov Inst. Math. 90 (1960).

Ya. G. Sinai, Introduction to Ergodic Theory (Princeton Univ. Press, Princeton, New Jersey, 1977).

L. Boltzmann, Ober die Beziehung eines allgemeine mechanischen Satzes zum zweiten Hauptsatze der Warmetheorie part II, Vienna, part II, 75, 67–73 (Sitzungsberichte Akad. Wiss., Vienna, 1877); English transl.: Stephen Brush, Kinetic Theory 2, p. 188.

A. Friedman, “Uber die Krummung des Raumes,” Zeitschrift fur Physik 10(1), 377–386 (1922).

A. D. Linde, Inflation and Quantum Cosmology (Acad. Press, Boston 1990).

Th. M. Nieuwenhuizen and I. V. Volovich, “Role of various entropies in the black hole information loss problem,” in Beyond the Quantum, eds. Th.M. Nieuwenhuizen, V. Spicka, B. Mehmani, M. J. Aghdami, and A. Yu. Khrennikov (World Scientific, 2007); [arXiv:hep-th/0507272].

H. Poincare, Remarks on the Kinetic Theory of Gases, Selected Works, Vol. 3 (Nauka, Moscow, 1974).

H. Poincare, (1904), “L’etat actuel et l’avenir de la physique mathematique,” Bulletin des sciences mathematiques 28(2), 302–324 (1904). English transl. in Poincare, Henri (1904), “The present and the future of mathematical physics,” Bull. Amer.Math. Soc. 37, 25–38 (2000).

V. V. Kozlov and D. V. Treschev, “Fine-grained and coarse-grained entropy in problems of statistical mechanics,” Theor. Math. Phys. 151(1), 120–137 (2007).

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge Univ. Press, New York).

I.V. Volovich, “Number theory as the ultimate physical theory,” p-Adic Numbers Ultr. Anal. Appl. 2(1), 77–87 (2010); Preprint No. TH 4781/87, CERN, Geneva, 1987.

I. V. Volovich, “p-Adic string,” Class. Quant. Grav. 4, L83–L87 (1987).

Yu. I. Manin, “Reflections on arithmetical physics,” in Mathematics as Metaphor: Selected Essays of Yuri I. Manin, pp. 149–155 (Amer. Math. Society, Providence, R.I., 2007).

V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994).

A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer Acad. Publishers, Dordreht, 1997).

B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Volovich, “On p-adicmathematical physics,” p-Adic Numbers Ultr. Anal. Appl. 1(1), 1–17 (2009); [arXiv:0904.4205 [math-ph]].

V. S. Varadarajan, “Multipliers for the symmetry groups of p-adic spacetime,” p-Adic Numbers Ultr. Anal. Appl. 1(1), 69–78 (2009).

E. I. Zelenov, “Quantum approximation theorem,” p-Adic Numbers Ultr. Anal. Appl. 1(1), 88–90 (2009).

I. V. Volovich, “Quantum cryptography in space and Bell’s theorem,” in Foundations of Probability and Physics, pp. 364–372, ed. A. Khrennikov (World Sci., 2001).

I. V. Volovich, “Seven principles of quantum mechanics,” [arXiv: quant-ph/0212126] (2002).

G.W. Mackey,Mathematical Foundations of Quantum Mechanics (W. A. Benjamin, New York, 1963).

L.D. Faddeev and O. A. Yakubovsky, Lectures on Quantum Mechanics for Mathematical Students, 2-nd edition (Moscow, 2001) [in Russian].

Yu. L. Klimontovich, Statistical Physics (Moscow, Nauka, 1982) [in Russian].

L. D. Landau and E.M. Lifshiz, Fluid Mechanics (Pergamon Press, 1959).

S. Flugge, Practical Quantum Mechanics (Springer, 1994).

W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH Verlag, 2001).

E. A. Dynin, “Statistical moments in quantum tunneling,” Soviet Physics, Semiconductors 24(44), 480–481 (1990).

I. V. Volovich and A. S. Trushechkin, “On quantum compressed states on interval and uncertainty relation for nanoscopic systems,” Proc. Steklov Math. Inst. 265, 1–31 (2009).

M. Reed and B. Simon, Methods of Modern Mathematical Physics 2 (Academic Press, 1975).

V. S. Vladimirov, Equations of Mathematical Physics (Marcel Dekker, New York, 1971).

A.N. Pechen and I. V. Volovich, “Quantum multipole noise and generalized quantum stochastic equations,” Infin. Dimen. Anal. Quant. Probab. Rel. Topics 5(4), 441–464 (2002).

V. P. Maslov and M. V. Fedoryuk, Semi-Classical Approximation in Quantum Mechanics (D. Reidel Publ., Dordrecht-Boston-London, 1981).

Yu. V. Prokhorov and Yu. A. Rosanov, Probability Theory (Springer-Verlag, Berlin, 1969).

A. V. Bulinsky and A. N. Shiryaev, Theory of Random Processes (Fizmatlit, Moscow, 2003).

A. Yu. Khrennikov, Interpretations of Probability (VSP Int. Publ., Utrecht, 1999).

M. Ohya and I. Volovich, Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems (Springer, 2011).